Distinct elastic properties and their origins in glasses and gels

The study addresses how glasses and gels—two non-equilibrium amorphous solids—differ in their elastic properties and how those properties evolve with observation time and aging. Glasses (metallic, polymeric, colloidal, granular) form by rapid quenching/densification and have homogeneous density beyond particle scales; gels are low-density, space-spanning networks formed by dynamical arrest of phase separation and exhibit very soft elasticity important in biology, food, pharmaceuticals, and cosmetics. Theories suggest that glasses of isotropic spherical particles self-organize structurally, gaining vibrational entropy at the expense of configurational entropy, whereas colloidal gels (with short-range reversible attractions) may arise via dense-phase glass transition or via percolation of local rigidity and hierarchical ordering driven by potential energy, indicating potentially distinct formation mechanisms (entropy-driven for glasses vs energy/interface-driven for gels). A central property is the elastic modulus, particularly the shear modulus G, which signals nonergodicity and solid-like response. For amorphous solids, the emergence of G is linked to mechanical force balance, mean-field theories, and relations to mean-squared displacements. Aging generally slows dynamics and reduces configurational space, suggesting increased G. The research question is to determine and contrast the observation-time and age dependence of shear and bulk moduli in glasses versus gels and to uncover the structural and dynamical mechanisms controlling these differences.

The paper reviews competing mechanisms of gelation: (1) glass transition within the dense phase during phase separation, linking gels and glasses as similar amorphous solids; (2) percolation of locally rigid structures and hierarchical ordering driven by potential energy, implying a fundamental difference from entropy-driven glasses. Prior work on shear modulus G in amorphous solids connects G to mechanical force balance (athermal/inherent states), mean-field theories at finite temperature, and to mean-squared displacement (MSD) scalings using density or infinite-frequency modulus. It also highlights aging-induced stiffening linked to decreasing configurational space. The review positions the present work to test whether these paradigms (e.g., G vs MSD scaling) hold similarly for gels and glasses, and to disentangle the roles of bulk versus interfacial free-energy reductions during aging.

Two simulation models were used. (1) Glasses: Molecular dynamics of a 2D binary mixture with harmonic pair potentials in NVT (Nosé-Hoover), N=4096, 1:1 large/small, diameter ratio 1.4, packing fraction 0.91. Units: ds=1 (small-particle diameter), mass m equal for both species; energy ε; time m ds^2/ε; temperature 10^-3 ε/kB. Known characteristic temperatures: Ton≈2.1, TMCT≈1.21, T0≈0.63. Protocol: equilibrate at T=3.0, quench to T=1.0 (< Tg), track aging time tw. 504 independent trajectories. (2) Gels: 3D polydisperse (Δd/d=0.04) colloids with Morse attraction u(rij)=ε exp[ρM(dij−rij)](exp[ρM(dij−rij)]−2), truncated/shifted at rcut=1.13 d. Parameters: N=10000, average diameter d, mass m, damping γ=0.1, timestep 0.002, units energy ε, time m d^2/ε, temperature ε/kB. Brownian time τB=(d/2)^2/(6D), with D=kBT/γ, used to scale tw. Conditions emulate PMMA colloid–polymer: φ=0.12, T=0.143, ρM=48.63 (from AO second virial for δ/d=0.13). Protocol: equilibrate at T=1.0 (> demixing Tc~0.3), quench to T=0.143 to form space-spanning networks. 23 trajectories. Elastic moduli measurements: For glasses (thermal state), elastic moduli computed via stress-fluctuation formalism: total modulus equals affine minus nonaffine contributions; affine term includes Born and pre-stress terms (kinetic negligible at low T); nonaffine term from stress fluctuations over a chosen observation time Δt. For gels, frequency-dependent storage moduli G' and K' obtained via oscillatory shear and volumetric deformation: apply sinusoidal strain ε(t)=ε0 sin(ωt), fit stress σ(t)=σ0 sin(ωt+δ)+σc to get storage (in-phase) and loss (out-of-phase) moduli; amplitudes verified to be in linear response via sweeps; focus on storage moduli G and K. Inherent (athermal) moduli: Obtain inherent structures by conjugate-gradient minimization to remove thermal fluctuations, then compute affine modulus as in thermal case (without kinetic term) and nonaffine modulus via Hessian matrix H=∂^2U/∂ri∂rj with affine force field; sum gives GIS and KIS. Dynamics and structural metrics: Mean-squared displacement (MSD). For 2D glasses, use cage-relative MSD (Δr^2)rel to remove Mermin-Wagner fluctuations; for gels, MSD of all particles and of isostatic particles (contact number Z≥6) to focus on the load-bearing network. Self-intermediate scattering functions Fc(k,Δt) used to locate plateau regimes. Structural measures: orientational order parameter Θ (deviation from ideal local packing), Voronoi-cell anisotropy β via Minkowski tensors, mean contact number Z, pressure p, potential energy per particle E and per contact 2E/Z, and network loop number Nloop from coarse-grained density-field binarization and Euler characteristic. Plateau modulus definition: use Δt=400 (glasses) and Δt/τB=343 (gels) to sample within vibrational plateau where particles undergo nonaffine vibrations without diffusion.

Observation-time dependence: In glasses, G(Δt) decreases from the affine/infinite-frequency value GA to a plateau (the vibrational plateau) and vanishes at long Δt when particles diffuse; K(Δt) drops rapidly from KA on a fast β timescale then remains essentially constant with further increases in Δt. In gels, both G and K decrease from their affine values to plateaus and eventually to zero at long Δt, showing similar dependence due to available configurational relaxation under both shear and compression in the presence of void space. Aging dependence (plateau moduli): Glasses show monotonic age-stiffening of G with tw, while K slightly decreases; the ratio G/K increases (decreasing Poisson’s ratio). Gels exhibit peak behavior in both G and K: initial stiffening with age followed by softening at later times; G/K also increases. Inherent vs thermal elasticity: In glasses, GIS and KIS are nearly constant with age, indicating that aging changes in thermal elasticity are dominated by thermal fluctuations/confinement rather than static structural changes. In gels, both GIS and KIS show peaks vs age, demonstrating that static structural evolution alone can generate the peak in thermal elasticity. Universal scaling with configurational constraints: For glasses, G/G∞ vs (Δr^2)−1 collapses as known, but this fails for gels. Instead, rescaling by inherent moduli yields robust collapses: M/MIS (with M=G or K) collapses against (Δr^2)−1 (using cage-relative MSD for glasses and the MSD of isostatic particles for gels), showing monotonic increase with aging. In gels, both G and K share the same configurational constraint, as K/KIS and G/GIS collapse together. Mechanisms and energetics: Total potential energy decrease can be decomposed into changes in contact number Z and contact energy 2E/Z. In glasses, over 90% of energy reduction comes from decreasing contact energy (optimizing existing contacts) with little change in Z; Θ decreases logarithmically with age (structural ordering), β decreases (more isotropic cages), dynamics slow, (Δr^2)−1 increases, G increases, while slight decreases in p and Z lead to a small decrease in K under constant volume. In gels, 2E/Z increases by ~10% on average while Z increases sufficiently that the overall energy still decreases (≈110% contribution from gaining contacts); Z grows and Nloop decreases linearly with log time, increasing configurational constraints and stiffening inherent structure via Z, but network-scale connectivity loss (lower Nloop) softens inherent elasticity later, producing a peak in GIS and KIS and thus in thermal M. Long-time limit distinction: At long observation times, glasses behave like liquids (finite K, zero G), while gels behave like gases (both K and G tend to zero). Representative quantitative choices: plateau evaluated at Δt=400 (glasses) and Δt/τB=343 (gels); in gels, MSD of isostatic particles (Z≥6) is required to reveal vibrational plateau and achieve scaling; in glasses, more than 90% of energy decrease is due to contact-energy reduction, while in gels, contact energy increases ~10% and energy decrease is driven by Z increase (~110% contribution).

The findings resolve how glasses and gels, despite both being amorphous and aging, differ fundamentally in their elastic responses due to distinct free-energy minimization routes: bulk free-energy driven structural ordering in glasses versus interfacial free-energy driven interface reduction in gels. In glasses, structural ordering (lower Θ and β) slows dynamics and enhances configurational constraints, elevating G while K is largely set by volumetric constraints (density/contact number) and thus changes little; constant-volume conditions explain the slight decrease in K. In gels, reduction of interfacial area increases particle-scale connectivity (Z) but reduces network-scale connectivity (Nloop), and both shear and bulk responses are controlled by similar configurational constraints because the network can dilate to relax compression in voided structures. This competition produces nonmonotonic aging (a peak) in both inherent and thermal moduli. The universal collapse of M/MIS versus inverse MSD demonstrates that thermal elasticity at finite temperature factorizes into a structural part (inherent elasticity) and a dynamical/configurational constraint part, with the latter quantifiable by appropriate MSDs (cage-relative in glasses; isostatic-particle and collective strand motion in gels). These insights explain disparate experimental observations (e.g., strong changes in G with modest changes in K across glass processing histories) and clarify why gels can lose both shear and bulk rigidity over long observation times, unlike glasses that retain finite K.

The study establishes distinct elastic behaviors of glasses and gels across observation time and aging, traces them to different free-energy minimization mechanisms (structural ordering vs interface reduction), and demonstrates that finite-temperature elastic moduli can be expressed as inherent moduli modulated by configurational constraints quantified by inverse MSD. Key contributions include: (1) identification of contrasting observation-time dependences (glasses: G decreases while K is nearly constant; gels: both G and K decrease similarly), (2) discovery of monotonic age-stiffening of G in glasses versus peak behavior in gels for both G and K, (3) demonstration that rescaling by inherent moduli yields universal collapses with inverse MSD, and (4) mechanistic link of aging to contact optimization in glasses versus contact gain and loop loss in gels. Future research directions include extending to 3D glass models and different interaction potentials, exploring pressure-controlled (isobaric) conditions to compare with experiments, quantifying collective strand dynamics in gels across length scales, and validating the factorization framework (M≈MIS×f((Δr^2)−1)) in experiments and other amorphous systems (e.g., polymers, metallic glasses, jammed packings).

The simulations employ idealized isotropic spherical particles and specific interaction models (harmonic for 2D glasses; Morse for 3D gels) and quench protocols, which may limit generality across materials. Glass simulations are 2D (chosen for long aging windows); while behavior is argued to be consistent with 3D, finite dimensionality may affect details. Constant-volume conditions in simulations differ from isobaric experimental conditions, influencing trends in bulk modulus K (e.g., simulated K decreases slightly with age in glasses, whereas experiments under constant pressure may show increases). Gels use a specific low volume fraction (φ=0.12) and interaction range (ρM tuned to AO equivalence), and only 23 independent trajectories were simulated, potentially limiting statistical sampling of network topology evolution. Scaling analyses for gels require focusing on isostatic particles, indicating that highly mobile interfacial particles and collective strand motions complicate direct MSD–modulus relations.